A285076 - OEIS

%I #24 Aug 26 2018 04:46:21

%S 1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,

%T 1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,

%U 1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,1,0,1

%N 1-limiting word of the morphism 0->10, 1-> 010.

%C The morphism 0->10, 1->010 has two limiting words.  If the number of iterations is even, the 0-word evolves from 0 -> 10 -> 01010 -> 100101001010 -> 01010100101001010100101001010; if the number of iterations is odd, the 1-word evolves from 0 -> 10 -> 01010 -> 100101001010, as in A285076.

%C From _Michel Dekking_, May 27 2017: (Start)

%C (a(n-1)) is the Sturmian sequence

%C     s(r,1-r) = (floor((n+1)r+1-r)-floor(nr+1-r))) with r = sqrt(2)-1.

%C Moreover, a = 10c, where c = A159684 is the Sturmian sequence

%C     s(r,r) = (floor((n+2)r)-floor((n+1)r))),

%C fixed point of the morphism gamma: 0 -> 01, 1 -> 010.

%C The morphism gamma is the time-reversal of the morphism psi: 0 -> 10, 1 -> 010 generating (a(n)).

%C This is a general property: see reference and link. It is also general that the square of psi has a second fixed point b = 01c, where b = A285073.  (End)

%H Clark Kimberling, <a href="/A285076/b285076.txt">Table of n, a(n) for n = 1..10000</a>

%H J. Berstel et P. Séébold, <a href="https://doi.org/10.1051/ita/1994283-402551">A remark on Sturmian words</a>, RAIRO - Theoretical Informatics and Applications 28 (1994), 255-263.

%H Michel Dekking, <a href="http://arxiv.org/abs/1705.08607">Substitution invariant Sturmian words and binary trees</a>, arXiv:1705.08607 [math.CO], 2017.

%H Michel Dekking, <a href="http://math.colgate.edu/~integers/sjs7/sjs7.Abstract.html">Substitution invariant Sturmian words and binary trees</a>, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.

%t s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {0, 1, 0}}] &, {0}, 13]; (* A285076 *)

%t Flatten[Position[s, 0]];  (* A285077 *)

%t Flatten[Position[s, 1]];  (* A285078 *)

%Y Cf. A285074, A285077, A285078.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Apr 19 2017